Brainercise Answers

Published Monday, March 7, 2011 by Tarang

Well, it seems that the brain teasers have stumped many of the best out there. Let us not keep the suspense up any longer and reveal the answers. I will restate the questions as well so that they are fresh in your minds:

1. You have a 3 gallon jug and a 5 gallon jug. You need to measure out exactly 4 gallons of water. How do you do it?

Answer: There are multiple ways of doing it, but here is the most obvious. Fill the 5 gallon jug completely with water. Then pour 3 gallons of water from the 5 gallon jug into the 3 gallon jug. So far, you have 2 gallons in the 5 gallon jug and the 3 gallon jug is completely full. Now empty out the 3 gallon jug. Pour the 2 gallons from the 5 gallon jug into the 3 gallon jug. The 5 gallon jug is empty now and the 3 gallon jug has 2 gallons. Now fill the 5 gallon jug completely with water. At this point, the 3 gallon jug has 2 gallons and the 5 gallon jug is full. Pour water from the 5 gallon jug into the 3 gallon jug until it is full. Remember the 3 gallon jug already had 2 gallons in it, so only 1 gallon can be poured into it from the 5 gallon jug. This leaves 4 gallons in the 5 gallon jug and you are done!

2. You have 4 stones of exact color, size, etc. except one of the stones is either heavier or lighter than the other 3. You are given a balance. How do you figure out which stone doesn’t belong in just two measurements of the balance?

Answer: Now you only know that one stone is a different weight but not if it is heavier or lighter.  Let us call the four stones A, B, C, and D. Weigh A against B. Scenario 1: A and B balance out equally. They are of equal weight, so C or D must be the different stone. Keep A on the balance and weigh it against C. If C and A are equal as well, then D is the odd one out. If A and C are not equal, then C is the odd one out. Scenario 2: A and B do not balance out equally. That means that either A or B is the odd one out. Keep A on the balance. Weigh it against C. If A and C are equal, then B must be the odd one out. If A and C are not equal, A is the odd one out. You are done! You may have noticed that you never actually had to weigh D at any point!

3. A rich merchant had collected many gold coins. He did not want anybody to know about them. One day, his wife asked, “How many gold coins do we have?”
After pausing a moment, he replied, “Well! If I divide the coins into two unequal numbers, then 32 times the difference between the two numbers equals the difference between the squares of the two numbers.”
The wife looked puzzled. Can you help the merchant’s wife by finding out how many gold coins they have?

Answer: This is the most mathematical of the brainercises. We can divide the coins into two unequal numbers, x and y. We can then convert his statement into an equation: 32(x-y)=(x^2)-(y^2). You may recall from algebra that (a+b)(a-b)=(a^2)-(b^2). So we know that (x+y)(x-y)=(x^2)-(y^2). By simple substitution, we know that x+y=32. So the total number of coins is 32.

4. You have 100 balls divided evenly into ten boxes (ten balls in each box). One of the boxes contains balls that are 1.5 grams lighter than the balls in the other boxes. You are given a weigh balance. How do you figure out which of the ten boxes contains the balls that are 1.5 grams less by using the balance. (Note: The boxes are numbered 1 through 10 and so are the balls from the boxes i.e. a ball from box 2 will be labeled 2)

Answer: My bad in wording the problem. I forgot to mention that it was a digital weigh scale. Try again now that you have the full problem!

5. In a far away land, it was known that if you drank poison, the only way to save yourself is to drink a stronger poison, which neutralizes the weaker poison. The king that ruled the land wanted to make sure that he possessed the strongest poison in the kingdom, in order to ensure his survival, in any situation. So the king called the kingdom’s pharmacist and the kingdom’s treasurer, he gave each a week to make the strongest poison. Then, each would drink the other one’s poison, then his own, and the one that will survive, will be the one that had the stronger poison.
The pharmacist went straight to work, but the treasurer knew he had no chance, for the pharmacist was much more experienced in this field, so instead, he made up a plan to survive and make sure the pharmacist dies. On the last day the pharmacist suddenly realized that the treasurer would know he had no chance, so he must have a plan. After a little thought, the pharmacist realized what the treasurer’s plan must be, and he concocted a counter plan, to make sure he survives and the treasurer dies. When the time came, the king summoned both of them. They drank the poisons as planned, and the treasurer died, the pharmacist survived, and the king didn’t get what he wanted.
What exactly happened there?

Answer: Let us get all the facts straight. Order of events: drink the other person’s poison, then your own, and then either survive or die. The pharmacist survived; thus, the pharmacist’s own poison cancelled out the treasurer’s poison. The treasurer died; thus, the treasurer’s own poison was stronger than the pharmacist’s poison. The king did not get what he wanted; thus, he did not get the stronger poison. We also know that the incentive to make the stronger poison is to survive because the one with the stronger poison will be able to cancel out the other’s poison. We also know that the treasurer knew that the pharmacist would win because he is a pharmacist and deals with poisons and such, so the treasurer devised a plan. He would make his poison (let’s call it T) the weakest possible, so he chose T to be water. If the pharmacist drank T (which is water), then his own poison P, he would die. The pharmacist figured out this plan and decided to make P water as well. However, the treasurer was not aware that P would just be water but he knew that T (his own poison) was definitely water. As a preemptive measure, the treasurer took a weak poison before the actual competition. Thus the treasurer took a weak poison, then the other person’s (P), then his own (T), and so he died because P and T were just water. The pharmacist took T, then P, and survived because all he had was water. The king, however, thought that P was the stronger poison because the pharmacist survived. Thus the king didn’t get what he wanted because he got P and P was just water.

6. I have three friends. Two play football, two play tennis and two play golf. The one who does not play golf does not play tennis, and the one who does not play tennis does not play football. Which games does each friend play?

Answer: This one is simple if you use proof language. From reasoning you know that “If A then B” then “If not B then not A.” So “the one who does not play golf does not play tennis” is the same as “the one who plays tennis plays golf.” Thus one friend plays tennis and golf. If you use the same logic on the second friend, then he/she plays football and tennis. We already have the two that play tennis, so the last friend must play golf and football.

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